We now show that \(\operatorname{\mathrm {Hilb}}_d(\mathbf {A}^n)\)  and \(\operatorname{\mathrm {Hilb}}_d(\mathbf {A}^n,0)\) 
are \(\mathbf {A}^1\) -connected over an infinite field \(k\) . This seems to be harder
for \(\operatorname{\mathrm {Hilb}}_d(\mathbf {A}^n,0)\) , because
(for \(d>1\) ) this space contains no smooth
subschemes of \(\mathbf {A}^n\) . When \(n\ge d\) , the \(\mathbf {A}^1\) -connectedness
of these Hilbert schemes
can be proved using the ideas of [1]},
but here we want the results for all \(n\)  and \(d\) .
